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Sobre la solución del Hamiltoniano de Tavis-Cummings en Óptica Cuántica: dinámica y estructura basada en la base colectivas de spines

Mario José Félix Rojas

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.

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Generalization of the Tavis–Cummings model for multi-level anharmonic systems: Insights on the second excitation manifold

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We present a fast, memory-efficient, unitarity-preserving numerical method beyond the rotating-wave approximation for the closed Tavis-Cummings model in which a multilevel spin system interacts with a cavity mode. This model can describe the interaction of an ensemble of spins with a cavity mode in which the spin frequency and other parameters are time-dependent. The method exploits the fact that, while the Tavis-Cummings model is not tri-diagonal, it can be brought into tri-diagonal form by a change of basis that can be implemented purely by re-indexing (permuting basis elements), which is a fast operation. By truncating the Fock basis of the cavity mode, the computational complexity of the method is linear in the total dimension of the coupled system, both in time and memory. The method can be employed to simulate any closed quantum system whose Hamiltonian terms can be brought into tri-diagonal form.

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The purpose of this article is to present a comprehensive study of the Tavis–Cummings model from a system-theoretic perspective. A typical form of the Tavis–Cummings model is composed of an ensemble of noninteracting two-level systems (TLSs) that are collectively coupled to a common cavity resonator. The associated quantum linear passive system is proposed, whose canonical form reveals typical features of the Tavis–Cummings model, including$\sqrt{N}$- scaling, dark states, bright states, single-excitation superradiant, and subradiant states. The passivity of this linear system is related to the vacuum Rabi mode splitting phenomenon in Tavis–Cummings systems. On the basis of the linear model, an analytic form is presented for the steady-state output state of the Tavis–Cummings model driven by a single-photon state. Master equations are used to study the excitation properties of the Tavis–Cummings model in the multiexcitation scenario. Finally, in terms of the transition matrix for a linear time-varying system, a computational framework is proposed for calculating the state of the Tavis–Cummings model, which is applicable to the multiexcitation case.

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The Tavis-Cummings model is a paradigmatic central-mode model in which a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties, and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity, and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we establish here the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and we study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates, and it illustrates how these models support a collective spin description despite the presence of disorder. Published by the American Physical Society 2024

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The Tavis-Cummings model for more than one qubit interacting with a common oscillator mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the frequencies of the qubits are much smaller than the oscillator frequency and the coupling strength is allowed to be ultrastrong. The application of the adiabatic approximation introduced by Irish et al. [Phys. Rev. B 72, 195410 (2005)] for a single-qubit system is extended to the multiqubit case. For a two-qubit system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intramanifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-qubit dynamics that are different from the single-qubit case, including calculations of qubit-qubit entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived.

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Generalization of the Tavis–Cummings model for multi-level anharmonic systems

J A Campos-Gonzalez-Angulo, R F Ribeiro, Joel Y. Yuen-Zhou

Abstract The interaction between anharmonic quantum emitters (e.g. molecular vibrations) and confined electromagnetic fields gives rise to quantum states with optical and chemical properties that are different from those of their precursors. The exploration of these properties has been typically constrained to the first excitation manifold, the harmonic approximation, ensembles of two-level systems [Tavis–Cummings (TC) model], or the anharmonic single-molecule case. The present work studies, for the first time, a collective ensemble of identical multi-level anharmonic emitters and their dipolar interaction with a photonic cavity mode, which is an exactly solvable many-body problem. The permutational properties of the system allow identifying symmetry classified submanifolds in the energy spectrum. Notably, in this approach, the number of particles, typically in the order of several millions, becomes only a parameter from the operational standpoint, and the size of the dimension of the matrices to diagonalize is independent of it. The formalism capabilities are illustrated by showing the energy spectrum structure, up to the third excitation manifold, and the calculation of the photon contents as a permutationally invariant quantity. Emphasis is placed on (a) the collective (superradiant) scalings of light–matter couplings and the various submanifolds of dark (subradiant) states with no counterpart in the single-molecule case, as well as (b) the delocalized modes containing more than one excitation per molecule with no equivalent in the TC model. We expect these findings to be applicable in the study of non-linear spectroscopy and chemistry of polaritons.

arXiv:2101.09475•New Journal of Physics•May 13, 2021•29 citations
We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.
Confined electromagnetic modes strongly couple to collective excitations in ensembles of quantum emitters, producing light–matter hybrid states known as polaritons. Under such conditions, the discrete multilevel spectrum of molecular systems offers an appealing playground for exploring multiphoton processes. This work contrasts predictions from the Tavis–Cummings model in which the material is a collection of two-level systems, with the implications of considering additional energy levels with harmonic and anharmonic structures. We discuss the exact eigenspectrum, up to the second excitation manifold, of an arbitrary number N of oscillators collectively coupled to a single cavity mode in the rotating-wave approximation. Elaborating on our group-theoretic approach [New J. Phys. 23, 063081 (2021)], we simplify the brute-force diagonalization of N2 × N2 Hamiltonians to the eigendecomposition of, at most, 4 × 4 matrices for arbitrary N. We thoroughly discuss the eigenstates and the consequences of weak and strong anharmonicities. Furthermore, we find resonant conditions between bipolaritons and anharmonic transitions where two-photon absorption can be enhanced. Finally, we conclude that energy shifts in the polaritonic states induced by anharmonicities become negligible for large N. Thus, calculations with a single or few quantum emitters qualitatively fail to represent the nonlinear optical response of the collective strong coupling regime. Our work highlights the rich physics of multilevel anharmonic systems coupled to cavities absent in standard models of quantum optics. We also provide concise tabulated expressions for eigenfrequencies and transition amplitudes, which should serve as a reference for future spectroscopic studies of molecular polaritons.
We present a fast, memory-efficient, unitarity-preserving numerical method beyond the rotating-wave approximation for the closed Tavis-Cummings model in which a multilevel spin system interacts with a cavity mode. This model can describe the interaction of an ensemble of spins with a cavity mode in which the spin frequency and other parameters are time-dependent. The method exploits the fact that, while the Tavis-Cummings model is not tri-diagonal, it can be brought into tri-diagonal form by a change of basis that can be implemented purely by re-indexing (permuting basis elements), which is a fast operation. By truncating the Fock basis of the cavity mode, the computational complexity of the method is linear in the total dimension of the coupled system, both in time and memory. The method can be employed to simulate any closed quantum system whose Hamiltonian terms can be brought into tri-diagonal form.
The purpose of this article is to present a comprehensive study of the Tavis–Cummings model from a system-theoretic perspective. A typical form of the Tavis–Cummings model is composed of an ensemble of noninteracting two-level systems (TLSs) that are collectively coupled to a common cavity resonator. The associated quantum linear passive system is proposed, whose canonical form reveals typical features of the Tavis–Cummings model, includingN\sqrt{N}N​- scaling, dark states, bright states, single-excitation superradiant, and subradiant states. The passivity of this linear system is related to the vacuum Rabi mode splitting phenomenon in Tavis–Cummings systems. On the basis of the linear model, an analytic form is presented for the steady-state output state of the Tavis–Cummings model driven by a single-photon state. Master equations are used to study the excitation properties of the Tavis–Cummings model in the multiexcitation scenario. Finally, in terms of the transition matrix for a linear time-varying system, a computational framework is proposed for calculating the state of the Tavis–Cummings model, which is applicable to the multiexcitation case.
The Tavis-Cummings model is a paradigmatic central-mode model in which a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties, and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity, and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we establish here the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and we study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates, and it illustrates how these models support a collective spin description despite the presence of disorder. Published by the American Physical Society 2024
The Tavis-Cummings model for more than one qubit interacting with a common oscillator mode is extended beyond the rotating wave approximation (RWA). We explore the parameter regime in which the frequencies of the qubits are much smaller than the oscillator frequency and the coupling strength is allowed to be ultrastrong. The application of the adiabatic approximation introduced by Irish et al. [Phys. Rev. B 72, 195410 (2005)] for a single-qubit system is extended to the multiqubit case. For a two-qubit system, we identify three-state manifolds of close-lying dressed energy levels and obtain results for the dynamics of intramanifold transitions that are incompatible with results from the familiar regime of the RWA. We exhibit features of two-qubit dynamics that are different from the single-qubit case, including calculations of qubit-qubit entanglement. Both number state and coherent state preparations are considered, and we derive analytical formulas that simplify the interpretation of numerical calculations. Expressions for individual collapse and revival signals of both population and entanglement are derived.
Abstract The interaction between anharmonic quantum emitters (e.g. molecular vibrations) and confined electromagnetic fields gives rise to quantum states with optical and chemical properties that are different from those of their precursors. The exploration of these properties has been typically constrained to the first excitation manifold, the harmonic approximation, ensembles of two-level systems [Tavis–Cummings (TC) model], or the anharmonic single-molecule case. The present work studies, for the first time, a collective ensemble of identical multi-level anharmonic emitters and their dipolar interaction with a photonic cavity mode, which is an exactly solvable many-body problem. The permutational properties of the system allow identifying symmetry classified submanifolds in the energy spectrum. Notably, in this approach, the number of particles, typically in the order of several millions, becomes only a parameter from the operational standpoint, and the size of the dimension of the matrices to diagonalize is independent of it. The formalism capabilities are illustrated by showing the energy spectrum structure, up to the third excitation manifold, and the calculation of the photon contents as a permutationally invariant quantity. Emphasis is placed on (a) the collective (superradiant) scalings of light–matter couplings and the various submanifolds of dark (subradiant) states with no counterpart in the single-molecule case, as well as (b) the delocalized modes containing more than one excitation per molecule with no equivalent in the TC model. We expect these findings to be applicable in the study of non-linear spectroscopy and chemistry of polaritons.